Firstly, we study the local cohomological functor. Next, we introduce the notion of
overcoherent arithmetic 𝒟-modules. We prove that unit-root F-isocrystals are
overcoherent and that overcoherence is stable by direct images, extraordinary inverse
images and local cohomological functors. Moreover, we obtain, using this stability, a
cohomological formula for L-functions associated to the dual complexes of overcoherent
complexes. It extends Étesse-Le Stum’s for F-overconvergent isocrystals.